# How To Calculate Power

Lucky and Spear, two 400 kg horses carrying two jockeys, 80 kg each, race each other on a 500 m track. They both start exactly at the same time and gallop as fast as they can. After a few seconds Lucky arrives at the finish line and, 5 seconds later, Spear arrives as well.

From a physicist’s perspective, both horses travelled the same distance, since they both started and ended in the same place following the same track. Nevertheless, as you have probably already guessed, Lucky’s average speed was higher than that of Spear. So, besides their velocity, what was different about both racers?

They certainly performed the same physical work, since they both carried the same mass (that of the jockey) along the same distance. But, in fact, it was Lucky’s greater power —the rate at which it performed said work— what allowed it to win the race. Let’s discover more about this interesting physical entity.

## How to calculate power

To calculate power first determine the work that is performed by a force. Then determine how long the work was performed for. Finally, divide the work in joules by the time in seconds to calculate the total power in watts.

## What is power

Energy can neither be created or destroyed. It transforms from one type to the other. For example, when you ride a bike, the force applied by your legs to the pedals is transferred to the wheels, making them turn. Said force exists due to the contraction and relaxation of your leg muscles, which is possible thanks to a cellular process in which some molecules are broken, releasing energy. This way, the chemical energy stored in your body cells is transformed into kinetic energy, or movement.

Another example is switching the lights on. When you do so, you allow current, which possesses electrical energy, to flow through a special type of device that transforms it into radiant energy, or light. At the same time, the electrical energy flowing through the wires usually comes from hydroelectric power plants, nuclear plants, solar panels or the burning of fossil fuels, which transform substances’ potential energies, the Sun’s radiant energy or some fuels’ chemical energies into usable electricity. Different types of processes can, therefore, transform one type of energy into another. This transformation can also be seen as an energy transfer, from one process to another.

In mechanics, for example, this energy transfer is related to the work performed by a force on a body. Let’s refresh our memory a bit: work is defined as the energy transferred by a force to a body in virtue of the displacement that is generated by it. If you are not sure about the definition of displacement, go ahead and read our article about it!

Imagine you want to move a 50 kg box around your house. If you push it hard enough you will accelerate it, meaning you will change its velocity from 0 (at rest) to some value depending on how big the force is. Now, let’s say you push the box 10 meters along a straight line. In this case, you would perform work on the box, which means you would have to transfer energy to it through the force you apply on it. This work is calculated as:

Where F is the magnitude of the force you apply and s the magnitude of the displacement generated by it. A more general expression to calculate the work performed by a force on a body, that applies when these vectors are not necessarily aligned with each other, is the following:

Where is the angle between the force and displacement vectors. In such a scenario, a part of the force would generate displacement, while another part would be counteracted by other forces. Only a net force acting on a body different from null can perform work. This is because when forces are counteracted, and the balance is zero, there is no net energy transfer to the body. Keep in mind, this does not mean the body does not absorb energy in other forms, for example, as deformation when being compressed by the force.

Let’s go back to our example: when pushing the box along 10 m, work is performed on it throughout the entire way. This is because you must counteract the effect of friction between the box and the floor continuously, so that it keeps moving. By the end, you would have transferred a certain amount of energy to the box. Now, if you decide to move the same box 20 m, you would have to transfer twice as much energy to it. In this case, the magnitude of the displacement is twice as big, so according to equation 1, the final work would also be twice as big.

Work is a good indicator of how much energy is transferred to a body in virtue of its linear movement. Nevertheless, it does not indicate how fast energy is being transferred. In our second example, if you push the box with the same force as in the first case, you would probably need twice as much time to move it 20 m. The total time needed increases because the box will accelerate the same amount as before, but the distance is larger. In this case, the rate at which energy is being transferred to the box remains the same. Although the total work is greater, it takes you more time to perform it. Take a look a this image to clarify both scenarios:

Now, if you want to move the box 10 m but faster than in our first example, you would have to apply a greater force than before in order to accelerate the box a greater amount and be able to cover this distance in less time. According to equation 1, this effectively increases work performed on the box by you. Furthermore, since the time needed to move the box along the 10 m was less, you transferred more energy in less time, meaning the rate of transfer was greater than before. Take a look at the following image to grasp this situation better:

Power is precisely the amount that tells us how fast energy is being transferred, and in the field of mechanics it is defined as the rate at which work is performed. This means, it is a measurement of the energy transferred per unit time. Mathematically, it is expressed as:

Where Pavg is the mean power, and W the work performed in time t. Since work has units of energy (joules), the units of power are joules/s, which are known as watts. Equation 3 expresses the mean value of power because its transfer rate is not necessarily constant. This means it can change as it is being performed, for which this more general expression, in terms of work and time differentials, is valid:

In the introduction to this article we used a horse race as an example of work performed over a period of time. Lucky and Spear certainly performed the same work at the end of the race. Nevertheless, it doesn’t seem fair to Lucky, since it arrived first to the finish line. This is exactly how power lets us compare both performances: in this context, we can say Lucky had a greater power than Spear, and thus converted energy into movement faster than its opponent.

## How to calculate mechanical power

In the context of mechanics power can be calculated depending on the type of motion generated by a force. In the case of linear motion, as in the examples discussed in the previous section, it is first required to determine the work that was performed by a force on the body of interest. This can be achieved using equation 2 if the displacement vector is known. If you want to learn how to calculate it, visit our article on this topic.

Once work is known, it is then required to determine how long it was performed for. This way, the resulting power is simply the ratio of the calculated work and the measured time, as shown by equation 3.

Power is commonly used to characterize engines. You have probably heard about a car engine having 150.000 watts of power or, more commonly, 200 horsepower. If you want to learn about this interesting power unit, go ahead and read the section titled “What is horsepower”. In this context, power is measured as the product of the torque () that generates the rotation of the engine and the resulting angular velocity ():

## What is electrical power

The concept of power also applies to the field of electricity. For example, electrons flowing through a wire possess energy, which in this case is referred to as electric energy. They can move thanks to the electrostatic force generated by an electric field, which exists thanks to a potential difference between both ends of the wire. This potential difference can be then understood as the work (per unit charge) performed by the electric field. Electrical power can be calculated through the following equation:

Where I is the current flowing through the wire, measured in amperes, and V the potential difference between both ends, measured in volts.

You may have seen that light bulbs in the supermarket usually present their power in watts. In this context, this refers to the amount of electrical energy consumed by the light bulb per unit time. So if one is labeled as a 60 W light bulb it means it will require around 60 joules per second to work, and will probably produce more light than a 40 W light bulb. The resulting current can then be determined through equation 5, as long as the operating voltage is known. This is usually around 220 V in Europe and 110 V in America.

## What is horsepower

You have probably seen the term horsepower (hp) in car specifications or, more broadly, in the context of machines that use engines. Horsepower is a unit of power derived from experiments performed by the famous engineer James Watt, thanks to whom we name power unit watts

By observing several draft horses in a coal mine, he determined they were capable of lifting 330 lbf of coal 100 feet in the air in 1 minute, resulting in a total work of 33.000 lbf-ft being performed per minute. This means, the average horse had a power of 746 W, which was useful to compare the animal’s performance to that of new machines, like the steam engine developed by Watt. In summary:

## Sources:

Hugh D. Young, Roger A. Freedman, University Physics, vol. 1. 12nd edition. Pearson. 2009.

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